Close packing of uniform size spheres

ABSTRACT

A method for attaining ultimate effeciency in the packing of uniform-sized spheres; which may be used to improve the buoyancy of flotation systems utilizing small hollow spheres.

United States Patent Mast et al.

[541 CLOSE PACKING OF UNIFORM SIZE SPHERES [72] Inventors: Phillip W. Mast, Raleigh, NC; Laurent A. Beaubien, Arlington, Va.

[73] Assignee: The United States of America as represented by the Secretary of the Navy [22] Filed: Dec. 12, 1969 [21] App1.No.: 884,526

[52] US. Cl ..9/8, 161/168 [51] Int. Cl. ..B63b 21/00 [58] Field of Search ..9/8; 229/22, 18; 206/65 R, 206/65 B, 46 FR; 161/168,162,161';

[56] References Cited UNITED STATES PATENTS 2,112,486 3/1938 Francis et al ..206/65 UX [15 1 3,703,012 [451 Nov. 21, 1972 2,151,066 3/1939 Anderson ..229/18 X 2,173,494 9/1939 Rous ..229/22 2,243,080 5/ 1941 Brogden ..229/22 X 2,806,509 9/1957 Bozzacco et a1 ..161/161 2,844,116 7/1958 Potter ..114/50 3,178,736 4/1965 Gross ..9/8 3,302,843 2/1967 Sheehan ..206/65 X 3,338,203 8/1967 Moore ..244/119 UX 3,359,657 12/1967 l-ledberg ..229/22 X Primary Examiner-Milton Buchrler Assistant ExaminerGregory W. OConnor Attorney-R. S. Sciascia, Arthur L. Branning, J. G. Murray and M. L. Crane 57 ABSTRACT A method for attaining ultimate effeciency in the packing of uniform-sized spheres; which may be used to improve the buoyancy of flotation systems utilizing small hollow spheres.

3 Claims, 5 Drawing Figures PATENTEDN I973 3.703.012

SHEET 1 UF 2 INVENTORS LAURENT A. BEAUB/EN PHILLIP Hf MAST sy fxfi GENT ATTORNEY PATENTED 2 3 7 03,012

sum '2 or 2 INVENTORS LAURENT A. BEAUB/EN PHILLIP W MAST CLOSE PACKING OF UNIFORM SIZE SPHERES BACKGROUND OF THE INVENTION This invention is directed to methods of pouring spheres into properly shaped forms (containers) in order to achieve a maximum packing fraction, and to methods of packing uniform size buoyant spheres such that maximum buoyancy may be obtained in composite flotation systems.

Heretofore, various types of buoyancy materials and systems have been used in carrying out oceanic research or for buoys and other flotation devices. Deep-sea exploration and research has brought about a need for new developments in flotation systems such as solid, buoyant materials and hollow structures capable of withstanding pressures at great depths.

Research has been conducted into forming buoyant structures which arelight in weight, of high strength, and resistant to water absorption. Such research has been reported in an article Syntactic Foams For Deep Sea Engineering Applications by Israel Resnick and Alexsander Macander published in Naval Research Reviews pages 6l7 May 1968. Syntactic foam is a composite, light-weight material consisting of a low density hollow microsphere filler embedded in a resin matrix. It has been pointed out in the above article that buoys made with a polyester resin including varying sized randomly placed hollow glass microspheres having a size of from 30 300 microns have a net buoyancy of 22 pounds per cu.ft. The article further points out that variations in density and strength of syntactic foam materials depend on the different grades of hollow glass microspheres used in the material, and also on the packing fraction of the glass sphere filler. An article directed to packing spheres is found in NRL Report 6904 Optimum Packing Of Hollow Spheres In.

Buoyancy Materials" by P. W. Mast, L. A. Beaubien and D. R. Mulville, dated June 30,1969.

Further, it is known in the prior art that glass spheres have been usedin plastic laminate compositions for high strength-weight ratios as required. The structures have been made by randomly packing the hollow spheres within the synthetic plastic with no particular interest placed in the buoyancy qualitiesof the material.

SUMMARY OF THE INVENTION This invention is directed to methods of close packing uniform size spheres and to the use of such methods to achieve maximum buoyancy in flotation systems utilizing small hollows spheres. One is concemed with a choice of containers which are of such a shape that spheres poured into them will automatically arrange themselves into a hexagonal close-packed array (the densest (74 percent) known form of packing). The containers which have been found to produce this close packed array are in the form of inverted regular (that is, equilateral) square and triangular pyramids and similar truncated pyramids. The shape of the square and triangular pyramids is such that they can be used to fill larger spaces without leaving any voids. This packing method can be applied to hollow buoyant spheres to achieve maximum efficiency in flotation systems using such spheres. The arrays of hollow spheres may be infused with a resin matrix which cures to form a solid composite system, or may be containerized to allow for a fluid or viscous matrix. The array including close-packed spheres may also be infiltrated with smaller buoyant spheres to further increase buoyancy prior to introduction of the matrix material.

Statement of the Objects It is therefore an object of the present invention to provide a practical method of packing uniform size spheres in a container such that there will be a minimum void space remaining, i.e. hexagonal close packing.

Another object is to make use of a container having a configuration which avoids the need of any means other than simple pouring for close packing spheres within the container.

Still another object is to provide a method for forming a flotation system which will have a maximum packing fraction of buoyant spheres, and therefore optimum system buoyancy.

Yet another object is to provide a method for pouring spheres into a container without requiring any vibrating means for aiding in aligning the spheres in a specific array.

.Still another object is to provide modules having close packed uniform size spheres such that the modules may be assembled side by side to form a larger continuous volume.

BRIEF DESCRIPTION OF THE DRAWING DESCRIPTION OF THE DRAWING Now referring to the drawings, there are shown by illustration in FIGS. 1 and 2 containers in the shape of a regular triangular pyramid 10 or tetrahedron and a regular square pyramid 11 or a half octahedron. Tetrahedrons and octahedrons are known to have equilateral tfaces or sides. Each of the containers are shown positioned on their apex end and containing layers of close packed spheres. FIG. 1 illustrates the triangular pyramidal form 10 tipped at an angle with respect to the surface wherein the spheres are being poured into the container. As shown, the spheres are directed toward the upper solid angle 13 so that the spheres run down over the top layers of spheres and build up from the lower face 14. Using this method of pouring, the spheres will automatically align themselves such that they are close packed, with each layer of spheres arranged in a triangular array and physically touching each other. As shown, a single sphere lies in the apex with the next row adjacent thereto having two spheres adjacent to each of the wall surfaces. Therefore, each of the spheres in one layer will nest within the voids in the layer directly under it, and the outer most rows of spheres of any layer will extend up over and above a plane through the diameter of the outermost spheres of the layer below it. It is further noted that the number of spheres adjacent to a wall, beginning at the apex, increases in number by one in each subsequent layer; that is, the apex has a single sphere, the next layer has two spheres along side each wall surface of the container, the next layer has three and so forth until the container is filled. Each of the layers in the tetrahedron as shown in FIG. 1 forms a triangular shape. Likewise, each layer in the half octahedron forms a square such that, beginning at the apex and moving toward the upper most end, each square will have an increase of one in the number of spheres along the wall surface. No tipping of the octahedron during pouring is required for achievement of close packing.

It has been pointed out above that the tetrahedron and half octahedron provide uniquely shaped forms for close packing spheres. Not only are these shapes unique for packing spheres therein, but they are also suitable for being assembled together as shown in FIGS. 3 and 4 to provide a plate assemblage of a plurality of such forms.

For maximum packing fraction of spheres, it is possible to eliminate many of the excess of voids by contouring the sides of the pyramidal forms. Such a contouring of the sides may be accomplished by a stepping arrangement of the sides such as illustrated in FIG. 3. Such contoured pyramids may be meshed to form a composite group. FIG. 3 illustrates a pyramidal section having contoured sides wherein a partial section has been meshed therewith and a third section is positioned adjacent thereto which illustrates how the contoured sides may be meshed.

FIG. 4 illustrates how several pyramidal shapes may be formed into one unit to form a plate. FIG. 4 further illustrates that single pyramidal shapes may be grouped to form other layer structures having many close packed spheres.

It can be seen from FIG. 4 that when pyramidal forms are arranged to form a plate, this plate has inclined edges and is actually the truncated section of a larger pyramid. In the case where large solid plates of arbitrary thickness and extent are a desired end product they may be quickly close packed by pouring the spheres into a truncated form such as shown in FIG. 5. For aiding in aligning the spheres, the bottom should be contoured. Such contouring of the bottom can consist of the simulation of the top surface of one layer of spheres or of any contouring which will cause the first layer of poured spheres to form a square or rectangular array. Therefore, once the spheres have been poured into a container such as a triangular, square, or rectangular pyridmal shape, a resin may be added and properly cured to provide a rigid high strength structure. The strength and density properties depend on the type of resin and the type of material used for the spheres in the fabrication technique.

The above discussion has set forth a method of pouring spheres into a specially shaped structure to provide the: least amount of void space and the addition of matrix provides a rigid high strength structure. Such structures having properly arranged buoyant spheres may be used to provide buoyant high strength, water resistant flotation systems. Such flotation systems may be formed by close packing hollow buoyant spheres within a particularly shaped container and then filling the voidswith a resin with subsequent curing thereof.

For greater buoyancy, it is possible to use a resin such as'described in the Naval Review article wherein the resin includes therein a low density hollow microsphere filler which has been embedded in the resin matrix. These spheres embedded in the resin matrix have a sufficiently small diameter such that the microspheres will easily flow into the voids between the close-packed spheres. Addition of the microsphere elements into the matrix will provide a composite material which has a greater buoyancy than that formed by spheres with only resin as the material which fills the voids. Therefore, it is recognized that a combination of closely packed spheres and a matrix resin including. microspheres therein will provide a buoyancy composite with a much greater buoyancy than that which has been available in the past.

Heretofore, it has been set forth that for the greatest buoyancy the spheres should be close-packed and in contact with each other to provide the greatest number of spheres within a contained space with the minimum void space therebetween. When one considers forces on these spheres which is brought by outside forces and wherein such resin bound structures as heretofore proposed are formed in areas which must withstand great pressure, it may be more advantageous to provide a structure which has close packed spheres wherein the spheres do not actually touch each other. Therefore, it may be advisable to provide a spacing between the close-packed spheres by providing each sphere with a thin coating of resin prior to packing. Therefore, if each of these spheres have a thin resinous coating thereon, the spheres themselves will not be in actual contact with each other but will therefore have a spacing between each of the spheres with the coating thereon actually in contact. Such a structure may withstand greater contact stresses and sympathetic implosions, especially for larger spheres.

In carrying out the invention for buoyant composites pyramidal shaped containers used to assemble the spheres in a close-packed-array may be made of the same material as the resin material. However, it is possible to provide forms within which the spheres are placed and the resin added with subsequent curing. Once the element has been cured, the form may be removed and the element will then be available for use as a buoyant composite. Obviously, if the outer container is removed, the element will be more buoyant than it would be if the container still surrounds the formed buoyant material. It will also be obvious to one skilled in the art that the spheres may be contained within a closed container without any matrix resin material added to the form. Likewise, the spheres may be surrounded in a closed chamber with other buoyant substances such as a liquid or solid which has a low specific gravity. Once the array of close-packed spheres has been formed, any desired solid or liquid may be used as a matrix. However, it has been determined that the resin matrix used aides in controlling or preventing the propagation of sympathetic implosions through the surrounding medium. Therefore, under certain conditions, it is advisable to have a matrix of a syntactic foam material, whereas for other uses, it may be advantageous to use a fluid or no matrix at all.

Obviously many modifications and variations of the present invention are possible in the light of the above teachings. It is therefore to be understood that within the scope of the appended claims the invention may be practiced otherwise than as specifically described.

What is claimed and desired to be secured by Letters Patent of the United States is:

1. A highly buoyant structure which comprises:

a regular pyramidal shaped structure,

said pyramidal shaped structure having equilateral faces,

a plurality of separate layers of close-packed uniform-sized hollow buoyant spheres arranged in said pyramidal structure with each layer including linear rows of side-by-side spheres with adjacent spheres nesting in voids formed by adjacent spheres with each sussessive layer shifted diagonally with respect to its adjacent layer,

a matrix of buoyant material filling the void spaces between said spheres,

said buoyant matrix material binding said shperes close-packing 5 pyramidal shaped body, which comprises:

positioning said equilateral pyramidal shaped body with its apex end in a downward position and its open end upwardly with the pyramidal center line tilted at an angle relative to a vertical line,

pouring a plurality of buoyant, hollow, uniform sized spheres into said open end of said body toward the upmost side of said body whereby said spheres align themselves in separate layers with successive layers including linear rows of side-by-side spheres with said spheres seated in voids formed by adjacent spheres, and

pouring a matrix of buoyant material over said spheres to fill all void spaces between said spheres and to bind adjacent spheres to each other.

3. A method as claimed in claim 2, which includes,

curing said matrix material to form a solid composite structure. 

1. A highly buoyant structure which comprises: a regular pyramidal shaped structure, said pyramidal shaped structure having equilateral faces, a plurality of separate layers of close-packed uniform-sized hollow buoyant spheres arranged in said pyramidal structure with each layer including linear rows of side-by-side spheres with adjacent spheres nesting in voids formed by adjacent spheres with each sussessive layer shifted diagonally with respect to its adjacent layer, a matrix of buoyant material filling the void spaces between said spheres, said buoyant matrix material binding said shperes together to form a structure of greater strength than a structure free of said matrix material.
 1. A highly buoyant structure which comprises: a regular pyramidal shaped structure, said pyramidal shaped structure having equilateral faces, a plurality of separate layers of close-packed uniform-sized hollow buoyant spheres arranged in said pyramidal structure with each layer including linear rows of side-by-side spheres with adjacent spheres nesting in voids formed by adjacent spheres with each sussessive layer shifted diagonally with respect to its adjacent layer, a matrix of buoyant material filling the void spaces between said spheres, said buoyant matrix material binding said shperes together to form a structure of greater strength than a structure free of said matrix material.
 2. A method of forming a composite structure by close-packing spheres into a hollow equilateral pyramidal shaped body, which comprises: positioning said equilateral pyramidal shaped body with its apex end in a downward position and its open end upwardly with the pyramidal center line tilted at an angle relative to a vertical line, pouring a plurality of buoyant, hollow, uniform sized spheres into said open end of said body toward the upmost side of said body whereby said spheres align themselves in separate layers with successive layers including linear rows of side-by-side spheres with said spheres seated in voids formed by adjacent spheres, and pouring a matrix of buoYant material over said spheres to fill all void spaces between said spheres and to bind adjacent spheres to each other. 